So basically I just get three mesh network routers that support an ethernet backbone, connect them to the ISP router via Ethernet (or to each other, and only the first to the ISP router?) and then I have one big network throughout the house?
Correct. And what you are proposing is the simplest and cheapest solution (apart from cheapest mesh using wireless backhaul only which give poor performance in a large house).
One big network is crucial to this working properly so you can walk around the house with your iPhone or iPad and stay connected. Each AP will handoff to the next one as signal drops off and signal from the nearest AP increases. At least in a perfect world that would happen.
Sidebar.
Getting handoff to work in real world is typical wireless technology.. sometimes brilliant, sometimes terrible. There's currently still a need for a voodoo priestess with lots of beads and rattles to placate the gods of the ether(net) who requires sacrifices of chicken livers to allow wifi to work. (wifi being a voodoo black art still)
That is why you want everything wired that can possibly be wired.. it is a mature technology. Not dependent on voodoo.
Apple sell one system in their stores, Linksys Velop, which has a review in Dong. (apple name it differently? to charge more??)
I think it is this one but do double check.
https://dongknows.com/linksys-velop-mx4200-ax4200-mesh-router-review/
All after sales support is from Linksys.. so buy from cheapest supplier and note they do have inbuilt setup which is hidden from average user.
https://dongknows.com/web-interface-vs-mobile-app/
Bad news about Time Capsule if you say it's that unreliable.
Again just for clarity.. Time Capsule is the old hardware router made by Apple up to 2018. It does not support SMB3 which is required now for security on all Mac, PC and mobile devices.
Time Machine (often confused with Time Capsule) is the backup software built into every Mac since Lion.
I have long recommended here people switch to Carbon Copy Cloner. It is reliable, inexpensive, and one license covers all the computers in your household.
Note: I have no pecuniary interest, no links to any seller and pay for license like everyone else.
Did I understand correctly that adding a HDD to a router usually does not work well, but the RT2600AC is one of the few that does work ok?
Yes, that is correct. The RT2600AC is fairly old now and still sells for well above better spec AX routers. It is not a good buy, and I wish Synology would refresh their product lineup. Or announce clearly it has scrapped the whole idea of selling routers.
The issue is mixing things up.
You mentioned in your first post using the ISP router.. that is fine as long as the mesh system you buy does NOT need to take over as main router. Not sure with the velop system so that is something else to check. Otherwise you are better buying a modem only and using your own router. The best way is to use the Velop since it then controls both the network and internet connection. The more you stick to one product from one brand the less issues you will have. (Unless you have Cisco certification!!)
Velop does support plugging in a hard drive and using it for storage.. purely for PC systems. It is useless for Mac backups. And is very likely SMB1 which is main vector used by ransomware.. just saying it is NOT something you want.
Sidebar
The Synology RT2600AC uses SMB2 but Apple computers as well as iOS require SMB3.
Backups do work but only by using AFP protocol which Apple is expected to drop soon. It is already deprecated.
All Synology NAS support SMB3 even base model which sells for about the same as the RT2600AC.. is far superior and will last several routers. Plus you can use it to copy files to and from iOS devices.
It is simple to plug any NAS by short ethernet patch cable into whatever main router you use or network switch.
So in the long run Synology NAS is the way to go.. whatever you buy as network routers and WAP.